Sum rules and bounds on the exchange and correlation energy functional of current-density functional theory

Abstract

The exact expression for the exchange-correlation energy functional ${E}_{\mathrm{xc}}[\ensuremath{\rho},{\mathbf{j}}_{p}]$ of the current-density functional theory (CDFT) is derived by means of the coupling-constant integration technique. It contains the coupling-constant-averaged pair correlation function, which is a functional of the electron density $\ensuremath{\rho}(\mathbf{r})$ and paramagnetic current density ${\mathbf{j}}_{p}(\mathbf{r}).$ On the basis of this expression, the local density approximation and its modifications for ${E}_{\mathrm{xc}}[\ensuremath{\rho},{\mathbf{j}}_{p}]$ are proposed within the CDFT. In addition, we present sum rules and bounds on ${E}_{\mathrm{xc}}[\ensuremath{\rho},{\mathbf{j}}_{p}]$ by considering the behaviors of the basic variables and ${E}_{\mathrm{xc}}[\ensuremath{\rho},{\mathbf{j}}_{p}]$ under the various types of the nonuniform coordinate scaling of electrons. They are useful in estimating the validity of the approximate forms of ${E}_{\mathrm{xc}}[\ensuremath{\rho},{\mathbf{j}}_{p}]$ proposed.